Are there real-world applications of toposes? Journalists would love to know, I guess. Toposes are fx used in program verification: e.g. https://url.au.m.mimecastprotect.com/s/WqzrC1WLjwsMpAlP9ILf9CVbcnO?domain=ir... This ranges from the topos of trees to more advantaged topos theory used in modal type theories or homotopy type theory.

On Mon, Sep 2, 2024 at 11:26 PM Patrik Eklund <peklund@cs.umu.se> wrote:

"Do we want to keep mathematics a dark secret, or what?"

Apparently, yes, we do.

Obviously, the definition of a topos is the same for all of us. But the way we use it, the way we attach it to other mathematical structures, is part of our own secrets. And there are apparently many such secrets.

I might even believe that Grothendieck's own perception, of what toposes really are, changed over time, and indeed in dialogue with the scientific community, a community which is not a closed one, but very much part of society.

Clearly, there may remain parts of "aus liebe zur Kunst" in topos theory, as for any part of mathematical theories for that matter, but generally speaking, there are always objectives, and requirements for theories to be applicable, applicability in a broader sense.

Are there real-world applications of toposes? Journalists would love to know, I guess.

Best,

Patrik

On 2024-09-02 08:32, Vaughan Pratt wrote:

"I'm relieved the journalist didn't try to explain what a topos was, or indeed anything mathematical."

Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what?

A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics.

For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs.

Vaughan Pratt

On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com> wrote:

Thanks - I saw this! I'm relieved the journalist didn't try to explain what a topos was, or indeed anything mathematical.

Sent

...

On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu> wrote:

An article about Alexander Grothiendieck has just appeared in the Guardian online newspaper. Be warned, it contains some seriously weird stuff! Toposes get a mention, though "not as we know them", along with Huawei, AI and Olivia Caramello. Beyond that, I'm not going to comment!

Since Microsoft mangles web addresses, here is the address with the punctuation removed:

www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence

Paul Taylor.

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Dear Vaughan, Of course you're right. But if the journalist had explained what a topos is, it might rather have destroyed his thesis that topos theory is a "magic bullet" which is going to solve all the problems of AI. I suppose it does no harm if people at Huawei believe that; and if it causes them to throw money at people doing research in topos theory, so much the better. But I remain sceptcal. I'm reminded of the occasion when Guerino Mazzola decided that topos theory was the "magic bullet" that would solve all the problems of musical analysis, and wrote a big book to prove his point: nothing much came of that in the long term. Peter Johnstone ________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu> Sent: 02 September 2024 06:32 To: Wesley Phoa <doctorwes@gmail.com> Cc: categories@mq.edu.au <categories@mq.edu.au> Subject: Re: Grothendieck in the Guardian "I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical." Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what? A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics. For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs. Vaughan Pratt On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> wrote: Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical. Sent

On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu<mailto:categories@paultaylor.eu>> wrote:

An article about Alexander Grothiendieck has just appeared in the Guardian online newspaper. Be warned, it contains some seriously weird stuff! Toposes get a mention, though "not as we know them", along with Huawei, AI and Olivia Caramello. Beyond that, I'm not going to comment!

Since Microsoft mangles web addresses, here is the address with the punctuation removed:

www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence

Paul Taylor.

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________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu> Sent: 02 September 2024 06:32 To: Wesley Phoa <doctorwes@gmail.com> Cc: categories@mq.edu.au <categories@mq.edu.au> Subject: Re: Grothendieck in the Guardian "I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical." Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what? A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics. For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs. Vaughan Pratt On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> wrote: Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical. Sent

On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu<mailto:categories@paultaylor.eu>> wrote:

An article about Alexander Grothiendieck has just appeared in the Guardian online newspaper. Be warned, it contains some seriously weird stuff! Toposes get a mention, though "not as we know them", along with Huawei, AI and Olivia Caramello. Beyond that, I'm not going to comment!

Since Microsoft mangles web addresses, here is the address with the punctuation removed:

www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence

Paul Taylor.

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Steve Vickers raises the issue of when a simplification is an over-simplification. One might think to view that as a function of the intended audience. For an audience as comfortable with the notion of a topological space as with the notions of sets and functions, I would agree that I'd oversimplified my account. Perhaps I was a little optimistic about mentioning graphs, but at least "graph" has no more syllables than "set", and these days half the student body is exposed to the concept thanks to Computer Science having recently passed Human Sexuality in enrollments. I had originally intended to mention the notion of an abelian category as being related to the notion of a topos. (Peter Freyd and I had a few exchanges on that topic in this forum a few decades ago that helped me a lot.) However I figured that abelian groups or vector spaces as typical of objects one might find in an abelian category were above the level of sets and functions, and so decided not to mention abelian categories. But we're all sufficiently fond of categories here to appreciate the value of being able to take the opposite of a category without feeling as ill as MSRI Director Bill Thurston claimed to be in his welcoming (and welcome) address to the Universal Algebra and Category Theory meeting at MSRI in July, 1993. What made Bill's remark memorable for me was the sharp intake of breath heard round the room when Bill made it. (And note how context can disambiguate a word like "welcome".) Does "function of the intended audience" have an opposite? Interestingly, it does. It is called the Gunning Fog Index for an article. It is calculated as 0.4 times the sum of the average sentence length and the percentage of three-syllable words or longer (with certain rules about counting syllables). The resulting number is the age group the article is optimum for. Unfortunately this does not take into account the level and type of education needed to know what a particular word means, and "topos" can only decrease the Fog Index. Likewise for philtrum, though acnestis will increase it, as words more likely to be encountered in Human Sexuality than Computer Science. But at least it's a start. A dictionary of obscure words with a score for each could be a blessing. And so one can just write at whatever level you feel is appropriate, and calculate its fog index when done. You then know what audience your exegesis is most suitable for. Vaughan Pratt On Mon, Sep 2, 2024 at 6:45 AM Steven Vickers <s.j.vickers.1@bham.ac.uk<mailto:s.j.vickers.1@bham.ac.uk>> wrote: Dear Vaughan, It's easy to make the summary "A topos is simply one of many possible generalization of sets and their functions ...", but that's definitely an over-simplification. As Mac Lane and Moerdijk say, toposes have two facets: as generalized universes of sets (which is what you said), and as generalized topological spaces (which is what I was alluding to in my own Guardian comment). In fact Johnstone's Elephant explicitly tries to bring out even more facets. The generalized topological spaces are at the heart of Grothendieck's motivation. On the one hand, that is in the sense of algebraic topology, in that topological invariants such as cohomologies can be calculated for them - and can be exploited in algebraic geometry. On the other hand, it is also in the sense of general topology, in that a topos can be fruitfully be viewed as a space whose points are the models of a geometric theory that the topos classifies. The trouble is, the generalized topological space is easy to lose sight of, even easier if you move to elementary toposes (which are not classifying toposes in their own right, but only relative to other toposes), so many mathematicians just see the generalized universes of sets. Actually, the "essential properties that make sets so valuable in mathematics" can be an obstruction to seeing the generalized topological spaces. The issue is that some of the "essential properties", such as cartesian closedness and subobject classifiers, do not interact successfully with geometric morphisms, the generalization of continuous maps. (They are not preserved by inverse image functors.) For an unobstructed view of the generalized topological spaces, at least in the sense of general topology, it is best to reject those non-geometric constructions. To summarize: if you view toposes as "simply" the generalized universes of sets, then you risk overlooking the generalized topological spaces. Steve. ________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu<mailto:pratt@cs.stanford.edu>> Sent: Monday, September 2, 2024 6:32 AM To: Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> Cc: categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto:categories@mq.edu.au>> Subject: Re: Grothendieck in the Guardian CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe. "I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical." Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what? A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics. For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs. Vaughan Pratt You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>

Dear Eduardo, I'm gratified to see your anecdote about Grothendieck being unhappy with "elementary topos". It confirms what I saw as a big problem with the current usage of "topos" when I wrote my article "Point-free generalized spaces, pointwise", https://url.au.m.mimecastprotect.com/s/qRDBCBNqgBC7VXLrZSzfkC2VWXi?domain=ar.... Regarding "Lawvere sets", Anel and Joyal have proposed the word "logos" for the categories of sheaves, the algebraic counterparts of the spatial notion of topos. Perhaps "Lawvere logos" is another candidate for "elementary topos"? Steve. ________________________________ From: Eduardo J. Dubuc <edubuc@dm.uba.ar> Sent: Tuesday, September 3, 2024 2:06 AM To: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; Vaughan Pratt <pratt@cs.stanford.edu> Cc: categories@mq.edu.au <categories@mq.edu.au> Subject: Re: Grothendieck in the Guardian CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe. Dear Steven, I very much agree with all in your posting, and I would like to add some comments: "A topos is simply one of many possible generalization of sets and their functions ... " is very misleading in fact as a topos Sets is a point. The underlying category of a topos shares the exactness properties of the category of sets, and only secondarily it is an elementary topos Morphisms of elementary topoi are the logical ones, and they sit in the other side of the duality, morphism of topoi (or their inverse images which go in the same direction than morphisms of elementary topoi) do no agree with the elementary topos essential structure of the underlying category. As Steven say, "The generalized topological spaces are at the heart of Grothendieck's motivation" I remember I was a student at Chicago when Grothendieck visited, and after his talk, around a table at a bar (he would drink only water) he said that he was very upset that Lawvere had called his concept a topos. I imagine he could have called "generalized set" for example. Much confusion would have being avoided, and today elementary topi would be called "Lawvere Sets", an even strong recognition of the importance of Lawvere and his creation. Eduardo. On 02/09/2024 10:45 AM, Steven Vickers wrote:

Dear Vaughan,

It's easy to make the summary "A topos is simply one of many possible generalization of sets and their functions ...", but that's definitely an over-simplification. As Mac Lane and Moerdijk say, toposes have two facets: as generalized universes of sets (which is what you said), and as generalized topological spaces (which is what I was alluding to in my own Guardian comment). In fact Johnstone's Elephant explicitly tries to bring out even more facets.

The generalized topological spaces are at the heart of Grothendieck's motivation. On the one hand, that is in the sense of algebraic topology, in that topological invariants such as cohomologies can be calculated for them - and can be exploited in algebraic geometry. On the other hand, it is also in the sense of general topology, in that a topos can be fruitfully be viewed as a space whose points are the models of a geometric theory that the topos classifies.

The trouble is, the generalized topological space is easy to lose sight of, even easier if you move to elementary toposes (which are not classifying toposes in their own right, but only relative to other toposes), so many mathematicians just see the generalized universes of sets.

Actually, the "essential properties that make sets so valuable in mathematics" can be an obstruction to seeing the generalized topological spaces. The issue is that some of the "essential properties", such as cartesian closedness and subobject classifiers, do not interact successfully with geometric morphisms, the generalization of continuous maps. (They are not preserved by inverse image functors.) For an unobstructed view of the generalized topological spaces, at least in the sense of general topology, it is best to reject those non-geometric constructions.

To summarize: if you view toposes as "simply" the generalized universes of sets, then you risk overlooking the generalized topological spaces.

Steve.

------------------------------------------------------------------------ *From:* Vaughan Pratt <pratt@cs.stanford.edu> *Sent:* Monday, September 2, 2024 6:32 AM *To:* Wesley Phoa <doctorwes@gmail.com> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian *CAUTION:* This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.

"I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical."

Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what?

A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics.

For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs.

Vaughan Pratt

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Dear Peter, I am aware of the recognition by Grothendieck of Lawvere contributions, specially concerning the subobject classifier, that he refereed to as "the Lawvere element" or rather "The Lawvere object", a term also used by Cartier. Carboni told me that he and Lawvere visited Grothendiek when he was already a hermit, he would not speak a word, but when he saw Lawvere he did!, and said "oh!, Bill !", and seemed very happy of Bill's visit. But only wrote things in pieces of paper as a way to communicate. Grothendieck's annoyance, in his Chicago visit, in an informal interchange at a table in Jimmy's bar in Hyde Park, was about to call or name "Topos" the Lawvere concept, not about that he had failed to spot the notion of subobject classifier. Grothendieck reflected a lot about which name to give to his main and fundamental concept, and came up with "Topos". He felt that when people say "Topos", they should be referring to his notion, and not to any other one. Eduardo Dubuc On 03/09/2024 1:48 PM, P.T. Johnstone wrote:

Dear Steve, dear Eduardo,

I can't take issue with Eduardo's anecdote, since I never met Grothendieck myself. But I have heard from other sources that his main unhappiness about elementary toposes was annoyance that he had failed to spot the notion of subobject classifier (which he referred to as "the Lawvere element") that made the elementary development possible. If he had, SGA4 might have looked very different!

And I don't think it's helpful to try to find another name for "elementary topos". The whole point of the story about the blind men and the elephant is that "wherever you touch it, it's still the same animal"; every topos, wherever it comes from (even ones like realizability toposes, whose origin is entirely logical) contains within itself both geometric and logical potentialities, and it's the interplay between these that gives the subject its power.

Incidentally, whilst logical functors are important, it's also important to remember that geometric morphisms (or at least their inverse image parts) are also "structure-preserving functors" in at least two senses (see A4.1.18 and B2.2.7 in the Elephant). So they would be natural morphisms to study even if one came to toposes from an entirely "logical" background.

Peter Johnstone

------------------------------------------------------------------------ *From:* Steven Vickers <s.j.vickers.1@bham.ac.uk> *Sent:* 03 September 2024 11:59 *To:* Eduardo J. Dubuc <edubuc@dm.uba.ar> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian Dear Eduardo,

I'm gratified to see your anecdote about Grothendieck being unhappy with "elementary topos". It confirms what I saw as a big problem with the current usage of "topos" when I wrote my article "Point-free generalized spaces, pointwise", https://url.au.m.mimecastprotect.com/s/hwUnCk815RCOnDj5LH2fOCGpvs0?domain=ar... .

Regarding "Lawvere sets", Anel and Joyal have proposed the word "logos" for the categories of sheaves, the algebraic counterparts of the spatial notion of topos. Perhaps "Lawvere logos" is another candidate for "elementary topos"?

Steve.

------------------------------------------------------------------------ *From:* Eduardo J. Dubuc <edubuc@dm.uba.ar> *Sent:* Tuesday, September 3, 2024 2:06 AM *To:* Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; Vaughan Pratt <pratt@cs.stanford.edu> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.

Dear Steven, I very much agree with all in your posting, and I would like to add some comments:

  "A topos is simply one of many possible generalization of sets and their functions ... "

is very misleading

in fact as a topos Sets is a point. The underlying category of a topos shares the exactness properties of the category of sets, and only secondarily it is an elementary topos

Morphisms of elementary topoi are the logical ones, and they sit in the other side of the duality, morphism of topoi (or their inverse images which go in the same direction than morphisms of elementary topoi) do no agree with the elementary topos essential structure of the underlying category.

As Steven say,

"The generalized topological spaces are at the heart of Grothendieck's motivation"

I remember I was a student at Chicago when Grothendieck visited, and after his talk, around a table at a bar (he would drink only water) he said that he was very upset that Lawvere had called his concept a topos.

I imagine he could have called "generalized set" for example.

Much confusion would have being  avoided, and today elementary topi would be called "Lawvere Sets", an even strong recognition of the importance of Lawvere and his creation.

Eduardo.

On 02/09/2024 10:45 AM, Steven Vickers wrote:

...

Dear Vaughan,

It's easy to make the summary "A topos is simply one of many possible generalization of sets and their functions ...", but that's definitely an over-simplification. As Mac Lane and Moerdijk say, toposes have two facets: as generalized universes of sets (which is what you said), and as generalized topological spaces (which is what I was alluding to in my own Guardian comment). In fact Johnstone's Elephant explicitly tries to bring out even more facets.

The generalized topological spaces are at the heart of Grothendieck's motivation. On the one hand, that is in the sense of algebraic topology, in that topological invariants such as cohomologies can be calculated for them - and can be exploited in algebraic geometry. On the other hand, it is also in the sense of general topology, in that a topos can be fruitfully be viewed as a space whose points are the models of a geometric theory that the topos classifies.

The trouble is, the generalized topological space is easy to lose sight of, even easier if you move to elementary toposes (which are not classifying toposes in their own right, but only relative to other toposes), so many mathematicians just see the generalized universes of sets.

Actually, the "essential properties that make sets so valuable in mathematics" can be an obstruction to seeing the generalized topological spaces. The issue is that some of the "essential properties", such as cartesian closedness and subobject classifiers, do not interact successfully with geometric morphisms, the generalization of continuous maps. (They are not preserved by inverse image functors.) For an unobstructed view of the generalized topological spaces, at least in the sense of general topology, it is best to reject those non-geometric constructions.

To summarize: if you view toposes as "simply" the generalized universes of sets, then you risk overlooking the generalized topological spaces.

Steve.

------------------------------------------------------------------------ *From:* Vaughan Pratt <pratt@cs.stanford.edu> *Sent:* Monday, September 2, 2024 6:32 AM *To:* Wesley Phoa <doctorwes@gmail.com> *Cc:* categories@mq.edu.au <categories@mq.edu.au> *Subject:* Re: Grothendieck in the Guardian *CAUTION:* This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.

"I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical."

Why would anyone object to journalists doing exactly those things?  Do we want to keep mathematics a dark secret, or what?

A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics.

For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set.  But not all properties, for example the law of the excluded middle, which holds for sets but not graphs.

Vaughan Pratt

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Camell Kachour has just let me know that 10 years ago he wrote to Myles Tierney. Myles kindly responded (21 Nov 2014) with: ================================== Dear Camell, Attached is the remark concerning Omega. It is from the first (ever) edition of "Cohomologie étale des schémas" by Artin and Grothendiek, 1963-1964. Best Myles =================================== I have put Myles' attachment at: http://science.mq.edu.au/~street/SGA.pdf Ross On 4 Sep 2024, at 11:51 AM, Ross Street <ross.street@mq.edu.au> wrote: On 4 Sep 2024, at 2:48 AM, P.T. Johnstone <ptj1000@cam.ac.uk> wrote: But I have heard from other sources that his main unhappiness about elementary toposes was annoyance that he had failed to spot the notion of subobject classifier (which he referred to as "the Lawvere element") that made the elementary development possible. If he had, SGA4 might have looked very different! Memory tells me that a subobject classifier does occur in SGA, perhaps in an example, with the \Omega notation which Lawvere and Tierney adopted. So Bill Lawvere was surprised when he heard Grothendieck's "Lawvere element" suggestion. Of course, the power that the cartesian closedness and the subobject classifier unleash was developed by Bill and Myles. Their's also was the ability to express a Grothendieck topology structure concisely as a monad on \Omega and to see it in Paul Cohen's forcing construction. Ross You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>

I suspect Wesley expected that a journalist trying to explain toposes (particularly as Grothendieck approached them) would have made a complete hash of it and left the public with misconceptions, rather than simply no conception. Best Thoughts, D.Y. ________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu> Sent: Monday, September 2, 2024 12:32 AM To: Wesley Phoa <doctorwes@gmail.com> Cc: categories@mq.edu.au <categories@mq.edu.au> Subject: Re: Grothendieck in the Guardian This email originated from outside of K-State. "I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical." Why would anyone object to journalists doing exactly those things? Do we want to keep mathematics a dark secret, or what? A topos is simply one of many possible generalization of sets and their functions that allows many other mathematical objects besides sets to be imbued with some of the essential properties that make sets so valuable in mathematics. For example graphs and their maps form a topos with very similar properties to sets and their functions, such as having the notion of a power set. But not all properties, for example the law of the excluded middle, which holds for sets but not graphs. Vaughan Pratt On Sun, Sep 1, 2024 at 1:16 PM Wesley Phoa <doctorwes@gmail.com<mailto:doctorwes@gmail.com>> wrote: Thanks - I saw this! I’m relieved the journalist didn’t try to explain what a topos was, or indeed anything mathematical. Sent

On Aug 31, 2024, at 1:13 PM, Paul Taylor <categories@paultaylor.eu<mailto:categories@paultaylor.eu>> wrote:

An article about Alexander Grothiendieck has just appeared in the Guardian online newspaper. Be warned, it contains some seriously weird stuff! Toposes get a mention, though "not as we know them", along with Huawei, AI and Olivia Caramello. Beyond that, I'm not going to comment!

Since Microsoft mangles web addresses, here is the address with the punctuation removed:

www theguardian com science article 2024 aug 31 alexander-grothendieck-huawei-ai-artificial-intelligence

Paul Taylor.

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